Harmonic Maass form

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Published on October 14, 2021 11:19:40 AM ● Video Link: https://www.youtube.com/watch?v=i9H5F1GkLTU

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In mathematics, a weak Maass form is a smooth function



f


{\displaystyle f}
on the upper half plane, transforming like a modular form under the action of the modular group, being an eigenfunction of the corresponding hyperbolic Laplace operator, and having at most linear exponential growth at the cusps. If the eigenvalue of



f


{\displaystyle f}
under the Laplacian is zero, then



f


{\displaystyle f}
is called a harmonic weak Maass form, or briefly a harmonic Maass form.
A weak Maass form which has actually moderate growth at the cusps is a classical Maass wave form.
The Fourier expansions of harmonic Maass forms often encode interesting combinatorial, arithmetic, or geometric generating functions. Regularized theta lifts of harmonic Maass forms can be used to construct Arakelov Green functions for special divisors on orthogonal Shimura varieties.

Source: https://en.wikipedia.org/wiki/Harmonic_Maass_form
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